Find The Derivative Of $ G(t) = \frac{t^2}{t-5} }$Choose The Correct Derivative Expression A. ${ G^{\prime (t) = \frac{(t-5)(2t) - T 2}{(t-5) 2} }$B. ${ G^{\prime}(t) = \frac{t^2 - (t-5)(2t)}{(t-5)^2} }$C. $[

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. When dealing with rational functions, finding the derivative can be a bit more challenging than with polynomial functions. In this article, we will explore how to find the derivative of a rational function using the quotient rule.

The Quotient Rule

The quotient rule is a fundamental rule in calculus that allows us to find the derivative of a rational function. It states that if we have a function of the form:

f(x)=g(x)h(x)f(x) = \frac{g(x)}{h(x)}

then the derivative of f(x)f(x) is given by:

f′(x)=h(x)g′(x)−g(x)h′(x)(h(x))2f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{(h(x))^2}

Applying the Quotient Rule to the Given Function

Now, let's apply the quotient rule to the given function:

g(t)=t2t−5g(t) = \frac{t^2}{t-5}

To find the derivative of g(t)g(t), we need to identify the numerator and denominator functions. In this case, we have:

g(t)=g(t)h(t)g(t) = \frac{g(t)}{h(t)}

where:

g(t)=t2g(t) = t^2

and

h(t)=t−5h(t) = t-5

Finding the Derivatives of the Numerator and Denominator

Now, let's find the derivatives of the numerator and denominator functions:

g′(t)=ddt(t2)=2tg^{\prime}(t) = \frac{d}{dt}(t^2) = 2t

and

h′(t)=ddt(t−5)=1h^{\prime}(t) = \frac{d}{dt}(t-5) = 1

Applying the Quotient Rule

Now that we have the derivatives of the numerator and denominator functions, we can apply the quotient rule to find the derivative of g(t)g(t):

g′(t)=h(t)g′(t)−g(t)h′(t)(h(t))2g^{\prime}(t) = \frac{h(t)g^{\prime}(t) - g(t)h^{\prime}(t)}{(h(t))^2}

Substituting the values of g(t)g(t), h(t)h(t), g′(t)g^{\prime}(t), and h′(t)h^{\prime}(t), we get:

g′(t)=(t−5)(2t)−t2(t−5)2g^{\prime}(t) = \frac{(t-5)(2t) - t^2}{(t-5)^2}

Evaluating the Answer Choices

Now, let's evaluate the answer choices to see which one matches the derivative we found:

A. ${ g^{\prime}(t) = \frac{(t-5)(2t) - t2}{(t-5)2} }$

B. ${ g^{\prime}(t) = \frac{t^2 - (t-5)(2t)}{(t-5)^2} }$

C. ${ g^{\prime}(t) = \frac{t^2 + (t-5)(2t)}{(t-5)^2} }$

Comparing the answer choices with the derivative we found, we can see that:

  • Answer choice A matches the derivative we found.
  • Answer choice B is the negative of the derivative we found.
  • Answer choice C is the sum of the derivative we found and the original function.

Conclusion

In this article, we explored how to find the derivative of a rational function using the quotient rule. We applied the quotient rule to the given function and found the derivative. We then evaluated the answer choices to see which one matched the derivative we found. The correct answer is:

A. ${ g^{\prime}(t) = \frac{(t-5)(2t) - t2}{(t-5)2} }$

Final Answer

The final answer is A\boxed{A}.

Discussion

This problem requires a good understanding of the quotient rule and how to apply it to rational functions. It also requires the ability to evaluate answer choices and choose the correct one. If you have any questions or need further clarification, please don't hesitate to ask.

Additional Resources

If you want to learn more about the quotient rule and how to apply it to rational functions, I recommend checking out the following resources:

  • Khan Academy: Quotient Rule
  • MIT OpenCourseWare: Calculus
  • Wolfram Alpha: Quotient Rule

Introduction

In our previous article, we explored how to find the derivative of a rational function using the quotient rule. We applied the quotient rule to a given function and found the derivative. In this article, we will answer some common questions related to finding the derivative of a rational function.

Q: What is the quotient rule?

A: The quotient rule is a fundamental rule in calculus that allows us to find the derivative of a rational function. It states that if we have a function of the form:

f(x)=g(x)h(x)f(x) = \frac{g(x)}{h(x)}

then the derivative of f(x)f(x) is given by:

f′(x)=h(x)g′(x)−g(x)h′(x)(h(x))2f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{(h(x))^2}

Q: How do I apply the quotient rule to a rational function?

A: To apply the quotient rule to a rational function, you need to identify the numerator and denominator functions. Then, you need to find the derivatives of the numerator and denominator functions. Finally, you can apply the quotient rule formula to find the derivative of the rational function.

Q: What are some common mistakes to avoid when applying the quotient rule?

A: Some common mistakes to avoid when applying the quotient rule include:

  • Forgetting to find the derivatives of the numerator and denominator functions
  • Not applying the quotient rule formula correctly
  • Not simplifying the derivative expression

Q: Can I use the quotient rule to find the derivative of a rational function with a negative exponent?

A: Yes, you can use the quotient rule to find the derivative of a rational function with a negative exponent. However, you need to be careful when simplifying the derivative expression.

Q: How do I evaluate answer choices when finding the derivative of a rational function?

A: When evaluating answer choices, you need to compare the derivative expression with the answer choices. If the derivative expression matches one of the answer choices, then that is the correct answer.

Q: What are some common answer choices for the derivative of a rational function?

A: Some common answer choices for the derivative of a rational function include:

  • The derivative expression with a positive exponent
  • The derivative expression with a negative exponent
  • The derivative expression with a fraction

Q: Can I use the quotient rule to find the derivative of a rational function with a variable in the denominator?

A: Yes, you can use the quotient rule to find the derivative of a rational function with a variable in the denominator. However, you need to be careful when simplifying the derivative expression.

Q: How do I simplify the derivative expression when finding the derivative of a rational function?

A: To simplify the derivative expression, you need to combine like terms and cancel out any common factors.

Q: What are some common mistakes to avoid when simplifying the derivative expression?

A: Some common mistakes to avoid when simplifying the derivative expression include:

  • Not combining like terms
  • Not canceling out common factors
  • Not simplifying the expression correctly

Conclusion

In this article, we answered some common questions related to finding the derivative of a rational function. We discussed the quotient rule, how to apply it, and some common mistakes to avoid. We also provided some tips for evaluating answer choices and simplifying the derivative expression. If you have any questions or need further clarification, please don't hesitate to ask.

Additional Resources

If you want to learn more about the quotient rule and how to apply it to rational functions, I recommend checking out the following resources:

  • Khan Academy: Quotient Rule
  • MIT OpenCourseWare: Calculus
  • Wolfram Alpha: Quotient Rule

I hope this article has been helpful in understanding how to find the derivative of a rational function using the quotient rule. If you have any questions or need further clarification, please don't hesitate to ask.